ABSTRACT
Classical references for the saddle-point methods are [5] and [15]. Let Ω be a bounded domain on Rn, S : Ω → R, f : Ω → R and λ > 0 be a large positive parameter. The Laplace method consists in studying the asymptotics as λ→∞ of the multi-dimensional Laplace integrals
F (λ) = ∫
f(x)e−λS(x)dx
Let S and f be smooth functions and we assume that the function S has a minimum only at one interior non-degenerate critical point x0 ∈ Ω:
∇xS(x0) = 0 , ∇2xS(x0) > 0 , x0 ∈ Ω
x0 is called the saddle-point. Then in the neighborhood of x0 the function S has the following Taylor expansion
S(x) = S(x0) + 1 2
(x− x0)†∇2xS(x0)(x− x0) + o((x− x0)3)
As λ→∞, the main contribution of the integral comes from the neighborhood of x0. Replacing the function f by its value at x0, we obtain a Gaussian integral where the integration over x can be performed
F (λ) ≈ f(x0)e−λS(x0) ∫
≈ f(x0)e−λS(x0) ∫ Rn e−
One gets the leading asymptotics of the integral as λ→∞
F (λ) ∼ f(x0)e−λS(x0) (
2pi λ
More generally, doing a Taylor expansion at the n-th order for S (resp. n− 2 order for f) around x = x0, we obtain
F (λ) ∼ e−λS(x0) (
2pi λ
akλ −k
k of the f and S at the point x0. For example, at the first-order (in one dimension), we find
F (λ) ∼ √
2pi λS′′(c)
e−λS(x0) (f(x0)
− 1 λ
( − f
′′(x0) 2S′′(x0)
+ f(x0)S(4)(x0)
8S′′(x0)2 + f ′(x0)S(3)(x0)
2S′′(x0)2 − 5 (S
′′′(x0)) 2 f(x0)
24S′′(x0)3
))